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21 August 1995

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PHYSICS

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ELSEVIER

LETTERS

A

Physics Letters A 204 (1995) 217-222

Amplification of atomic fields by stimulated emission of atoms * Ch.J. Bordk kboraruire de Physique des Lasers, URA du CNRS 282, Universife’ Paris-Nerd. Villetaneuse. France Laboratoire de Gravitation et Cosmologie Relativisres. URA du CNRS 769, Universitk Pierre et Marie Curie, Paris, France Received 22 June 1995; accepted for publication Communicated by JJ? Vigier

30 June 1995

Abstract We discuss the possibility of amplifying matter-waves by stimulated emission of massive bosons. Specifically, we consider the induced dissociation of molecules under the influence of an incident atomic wave and compare this to the special case of the induced emission of ground state atoms from excited atoms. Genera1 formulas for the gain and for the spontaneous dissociation rate. which apply to both cases, are derived from relativistic quantum field theory.

1. Introduction

The recent development of atom optics and interferometry and especially of atomic resonators has demonstrated that one could often exchange the roles of atoms and photons in optical devices in order to design an atomic wave device. This is part of a genera1 view of the world of photons and atoms as elementary field particles which can mutually scatter each other in a reciprocal way. For example, one could imagine to achieve a device equivalent to optical lasers where photons are replaced by integer spin atoms and where the atoms are reflected or trapped by light (“atom laser”, “atomaser” or “ataser”). One way towards this goal, which is currently being explored, is to achieve Bose-Einstein condensation with a collection of cold atoms in thermal equilibrium. We present an alternative approach, closer to the optical case, in which atoms are produced in the same phase space cell by the induced version of a spontaneous process, very far from thermal equilibrium. This process should be the analog of the spontaneous emission of photons,

(1)

A*-+ A + photon. If we replace photons by bosonic atoms B we are led to consider the more general dissociation C-+A+B,

process, (2)

*The results of this paper were first presented at the “Colloquium on the developments and perspectives of atomic interferometry” organized in honour of Professor A. Zeilinger by Professor C. Cohen-Tannoudji at the Fondation Hugot du College de France, Paris, on 5 April 1995. Elsevier Science B.V. SSDI 0375-9601(95)00466-l

218

Ch.J. Bord6 /Ph.wics

Letters A 204 (199.5) 217-222

which includes the emission of a photon A as a special case. In the case of an ordinary laser, the photon energy is provided to the emitter by an external energy source, which excites the atoms. Similarly, in the case of the atomic wave, the energy represented by the mass of the atomic constituents needs also to be provided to the emitters. If we do not want to have to recreate each of these masses individually from other particles, they must also be constituents of the emitter. The simplest solution is therefore to use for the emitter C a composite object (molecule, excited atom or ion) which includes the required boson B as a constituent.

2. Theoretical

framework

Like stimulated emission of photons, the enhancement factor in the emission of atoms in the same phase cell by an incoming atomic wave, comes only from the bosonic character and the exchange properties of indistinguishable particles. Following Feynman [ I 1, for II incident atoms B, the statistical weight of the initial state is l/n!, the statistical weight of the final state is l/(n + 1) !, and the amplitude for the process is (II + 1 )! times the amplitude for spontaneous emission. The probability (per second) of emission is then [ (II + 1) !] */( II + 1) !n! = n + 1 times the probability of spontaneous dissociation. This can be demonstrated rigorously in a quantized field description of the stimulated dissociation process. In such a description, atomic systems in a given internal state are described as elementary particles by quantum field operators [ 23 satisfying commutation or anti-commutation rules to introduce their bosonic or fermionic character. These quantum fields operate on a Fock space restricted to the positive energy particles sector (we exclude the formation of antiatoms or anti-molecules from atoms and molecules) and satisfy free-field equations. To take the atomic angular momentum into account, it is necessary to introduce fields for arbitrary spin written as Lorentz spinors or tensors. Here for simplicity, we shall assume scalar (pseudo-scalar) fields which satisfy the Klein-Gordon equation, corresponding to spin zero particles,

where (4) is the relativistic energy of the atom in state cy and where E,/c* corresponds to an expansion in plane waves. For many-particle discrete modes ~!a,,~ ( X) such that

The starting point is then simply an effective Hamiltonian UBE~

dr

d3pc

E2 C

Ec(Pc)EA(Pc-PB)

0

x exp{i[Ec@)

- E~(pc

-PB)

-

E~(Pn)lr/h-

is the zero-order population of species C, with a normalized where nc(0)~(pc) of width Mou. The coupling constant can be eliminated in favor of the decay constant, 1 & fi2 ~EB(P&BE; Like

in the case

broadened,

=-- h2 PBPBO of

depending

(15)

rr/2)E(Pc), momentum

distribution

E(pc)

rco 4~'

amplification, the gain may thus be either homogeneously or inhomogeneously on the relative size of W and of the inhomogeneous energy width. This inhomogeneous

light

221

Ch.J. Bordk / Physics Letters A 204 (1995) 217-222

width is of the order of the spread in kinetic energies (- Mcu2) in the case of massive particles but it is much larger (N Moue) when MA = 0. We will give expressions for the gain in the limits of a purely homogeneously or inhomogeneously broadened line shape. In the inhomogeneous case the r integral gives a 6 function for the energy and if the distribution F(pc) is a non-relativistic Maxwell-Boltzmann distribution of width Mou we obtain n&O’ = ----~*,/%~exp(c2/u2)(erf{(c/u)[E~(pBa)~~(~B) 4%-312pipe0 11

a(/‘B)

-erf{(c/u)[EB(PBo)EB(PB)

+PBCPBC~~/@)

(16)

-PBOPBC~~/-%}),

which gives again the usual Doppler profile when MB vanishes. When the boson mass is different the previous formula simplifies in the non-relativistic approximation,

from zero,

)I .

Both (16) and (17) exhibit a resonance extract the Einstein B coefficient,

(17)

for pB 2 pBB, if pro is large enough compared

to MBU. We can also

(18)

B, = h*crco/hp&?$

associated

with a line shape normalized

in frequency

v = pBC/h.

If MBU is large compared

to pB0 then

(19) Let us point out that, from ( 12) and ( 15)) one can show that, in the non-relativistic should be satisfied between the gain and the dissociation constant,

limit, the following

relation

00 4rr

uB&(pB)+%i

=%&"h3ko

s 0

and one can check that this is indeed satisfied by ( 17). This relation is the analog of the Fiichtbauer-Ladenburg formula in optical spectroscopy. The homogeneous gain formula is obtained in the limit where F(pc) is considered as a S function in (15), a(pB)

h2

= n;“-

EC

I

rrco/4

~BPBoEA(--PB)G(~/~)~+

[&-EA(-PB)

For MB = 0 we recover the usual formula for light amplification a(w)

=

n$g-

1 23r (A/2)2

with the natural width for r = Too = A,

(N2>2

(21)

+ (w - wo)2

(the factor 2 in the denominator comes from a proper account of the polarization homogeneous limit is obtained only in the extreme case fir > Mcuc, (0) 2 c ’ a(pB) =nc A”zG(A/2)2+

(20)

- EB(PB)I~/~~~’

(A/2>*

[Ec-EB(P,) -p~c]~/!i~'

of light).

For MA = 0,the

(22)

222

Ch.J. Bord6 / Physics Letters A 204 (1995) 217-222

For MA,MB,Mc &,B)

=

# O,r=Tca,

h’

n;” ---

(m2

h’fB 1

PBPBO GAB TT=(r/2j2

+ [(Pi,

-

(23)

P~)/%ABI~/~~~~

We see that, at resonance, the cross section is again proportional to a de Broglie wavelength squared, and therefore one should use atoms as slow as possible. The mass defect AE/c2 should thus be as small as possible.

5. Conclusion

and possible gain media

We get formulas for atomic waves which are similar to the optical case. This demonstrates that comparable gains could be obtained with the same wavelengths and population inversion densities. It should be noted, however, that a divergent gain occurs for vanishing atomic velocity and this could be in favor of the slowest atomic waves for a given length of amplifier and fixed resonator losses. In Eq. (1) species B is an atomic or a molecular boson, but A can be anything from a photon to a solid, i.e., C can be an excited atom, a negative ion, a molecule, a cluster or even a surface (B may in fact be trapped in any external potential). Especially interesting possibilities are the predissociation of electronically excited molecules (e.g. Na2 (B’II, ) +Na( 3p) +Na( 3s) ) or of vibrationally excited van der Waals molecules (e.g. He-12) or the dissociative recombination of ions (e.g. HeH+ + e- +HeH-He+H). The only requirement to maintain the population inversion is that species A should escape or be removed as fast as possible through spontaneous decay, optical pumping or any other means. Finally losses by elastic and inelastic collisions of B with A or C need to be investigated in each case.

Acknowledgement The author expresses many thanks to Drs. J.-C. Houard, Letokhov and M. Broyer for stimulating discussions.

J. Dalibard,

J.-Y. Courtois,

M.-C. Castex,

VS.

References [ I ] R.P. Feynman, Quantum electrodynamics, Frontiers in physics (Addison-Wesley, Reading, 1962). 121Ch.J. BordC, Density matrix equations and diagrams for high resolution non-linear laser spectroscopy, application to Ramsey fringes in the optical domain, in: Advances in laser spectroscopy, eds. ET. Arecchi, E Strumia and H. Walther (Plenum, New York, 1983).